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Theorem r19.3rm 3454
Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)
Hypothesis
Ref Expression
r19.3rm.1 𝑥𝜑
Assertion
Ref Expression
r19.3rm (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem r19.3rm
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2203 . . 3 (𝑎 = 𝑦 → (𝑎𝐴𝑦𝐴))
21cbvexv 1891 . 2 (∃𝑎 𝑎𝐴 ↔ ∃𝑦 𝑦𝐴)
3 eleq1 2203 . . . 4 (𝑎 = 𝑥 → (𝑎𝐴𝑥𝐴))
43cbvexv 1891 . . 3 (∃𝑎 𝑎𝐴 ↔ ∃𝑥 𝑥𝐴)
5 biimt 240 . . . 4 (∃𝑥 𝑥𝐴 → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
6 df-ral 2422 . . . . 5 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
7 r19.3rm.1 . . . . . 6 𝑥𝜑
8719.23 1657 . . . . 5 (∀𝑥(𝑥𝐴𝜑) ↔ (∃𝑥 𝑥𝐴𝜑))
96, 8bitri 183 . . . 4 (∀𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴𝜑))
105, 9syl6bbr 197 . . 3 (∃𝑥 𝑥𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
114, 10sylbi 120 . 2 (∃𝑎 𝑎𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
122, 11sylbir 134 1 (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1330  wnf 1437  wex 1469  wcel 1481  wral 2417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-cleq 2133  df-clel 2136  df-ral 2422
This theorem is referenced by:  r19.28m  3455  r19.3rmv  3456  r19.27m  3461  indstr  9413
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